CSE 331 Introduction to Algorithm Analysis and Design. Sample Mid-term Exam-I: Fall 2018

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1 NAME: CSE 331 Introduction to Algorithm Analysis and Design Sample Mid-term Exam-I: Fall 2018 Atri Rudra DIRECTIONS: Closed Book, Closed Notes except for one review sheet. Time Limit: 50 minutes. Answer the problems on the exam paper. Make sure you write your NAME on the paper. If you need extra space use the back of a page. 1a /5 1b /5 1c /5 1d /5 1e /5 1f /5 1g /5 1h /5 Total /40 FEW GENTLE REMINDERS: You can quote any result that we covered in class or any problem that was there in a homework or recitation (but remember to explicitly state where you are quoting a result from). If you get stuck on some problem for a long time, move on to the next one. The ordering of the problems is somewhat related to their relative difficulty. However, the order might be different for you! You might be better off by first reading all questions and answering them in the order of what you think is the easiest to the hardest problem. Keep the points distribution in mind when deciding how much time to spend on each problem. 1

2 1. (8 5 = 40 points) Each of the questions below have two parts. For the first part, you need to give a justification for the statement and is worth 2 points. For the second part, answer True or False and briefly A correct answer with no or totally incorrect justification will get you 1 out of the total 3 points. An incorrect answer irrespective of the justification will get you 0 out of 3 points. You can assume part 1 when answering part 2 but to get credit for part 1, you have to answer part 1. (Recall that a statement is true only if it is logically true in all cases while it is is false if it is not true in some case). (a) Consider an arbitrary instance of the stable marriage problem with n men and n women. There are n! = n (n 1)... 1 many possible perfect matchings. There are at most n! stable matchings for the instance. (b) Let f(n) = log log n and g(n) = f(n) is Ω(g(n)). f(n) is O(g(n)). 2

3 (c) Let f(n) = n n and g(n) = 2 400n. f(n) = 2 n log 2 n. f(n) is Ω(g(n)). (d) Let a 1,..., a n be n integers. Let a i {0, 1} for each i [n]. Then the n numbers can be sorted in O(n) time. Let n a i n for every i [n]. Then the n integers a 1,..., a n can be sorted in O(n) time. 3

4 (e) Consider the BFS algorithm with its input graph G in adjacency list format. The input size for BFS is Θ(n + m). BFS is a linear time algorithm. (Recall that an algorithm is a linear time algorithm if it runs in time O(N) on inputs of size N.) (f) For any graph, recall that running the BFS algorithm implicitly computes a BFS tree. (Note: BFS tree is not rooted.) A BFS tree can be computed (explicitly) in O(m + n) time. Every graph has a a unique BFS tree for it. 4

5 (g) Recall that a directed graph is strongly connected if and only if every pair of vertices have directed paths from one to the other. A directed graph has at most n 2 edges in it. Any directed graph on n vertices with at least n 1 edges is strongly connected. (h) Recall that any graph can be represented in adjacency matrix format. Adjacency matrix takes Θ(n 2 ) space. There is an O(n 2 ) time algorithm that for any graph on n vertices given in its adjacency matrix, converts it into its adjacency list representations. 5